ABSTRACT
Different representations are used to encode the geometry of three-dimensional objects (see Chapter 1). The choice of a representation depends on the acquisition process upstream and on the application downstream. However, the representations that are the easiest to reconstruct are in most cases not optimal for the applications. The notion of parameterization attaches a “geometric coordinate system” to the object (see Chapter 3). This chapter introduces methods that compute such a parametric representation for a given polygonal mesh. This facilitates converting from one representation to another. For instance, it is possible to convert a mesh model into a piecewise bicubic spline surface, which is the type of representation used in computer-aided design (CAD) packages. In a certain sense, this retrieves an “equation” of the geometry, or constructs an abstraction of the geometry: once the geometry is abstracted, re-instancing it into alternative representations becomes easier.
